A generalization of supplemented modules

Abstract

Let \(R\) be an arbitrary ring with identity and \(M\) a right \(R\)-module. In this paper, we introduce a class of modules which is an analogous of  \(\delta\)-supplemented modules defined by Kosan. The module \(M\) is called  principally \(\delta\)-supplemented, for all \(m\in M\) there exists a submodule \(A\) of \(M\) with \(M = mR + A\) and \((mR)\cap A\) \(\delta\)-small in \(A\). We prove that some results of \(\delta\)-supplemented modules can be extended to principally \(\delta\)-supplemented modules for this general settings. We supply some examples showing that there are principally \(\delta\)-supplemented modules but not \(\delta\)-supplemented. We also introduce principally \(\delta\)-semiperfect modules as a generalization of \(\delta\)-semiperfect modules and investigate their properties.